HMMT November

Number theory Difficulty 4.5 AIME Find the answer

How many perfect squares divide $10^{10}$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

A perfect square $s$ divides $10^{10}$ if and only if $s=2^{a} \cdot 5^{b}$ where $a, b \in\{0,2,4,6,8,10\}$ . There are 36 choices, giving 36 different $s$ 's.

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.